Sinusoidal spiral

Sinusoidal spirals: equilateral hyperbola (n = -2), line (n = -1), parabola (n = -1/2), cardioid (n = 1/2), circle (n = 1) and lemniscate of Bernoulli (n = 2), where rⁿ = 1ⁿ cos(nθ) in polar coordinates and their equivalents in rectangular coordinates.

In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates

$r^n = a^n \cos(n \theta)\,$

where a is a nonzero constant and n is a rational number other than 0. With a rotation about the origin, this can also be written

$r^n = a^n \sin(n \theta).\,$

The term "spiral" is a misnomer, because they are not actually spirals, and often have a flower-like shape. Many well known curves are sinusoidal spirals including:

• Equilateral hyperbola (n = −2)
• Line (n = −1)
• Parabola (n = −1/2)
• Tschirnhausen cubic (n = −1/3)
• Cayley's sextet (n = 1/3)
• Cardioid (n = 1/2)
• Circle (n = 1)
• Lemniscate of Bernoulli (n = 2)

The curves were first studied by Colin Maclaurin.

Equations

Differentiating

$r^n = a^n \cos(n \theta)\,$

and eliminating a produces a differential equation for r and θ:

$\frac{dr}{d\theta}\cos n\theta + r\sin n\theta =0$.

Then

$\left(\frac{dr}{ds},\ r\frac{d\theta}{ds}\right)\cos n\theta \frac{ds}{d\theta} = \left(-r\sin n\theta ,\ r \cos n\theta \right) = r\left(-\sin n\theta ,\ \cos n\theta \right)$

which implies that the polar tangential angle is

$\psi = n\theta \pm \pi/2$

and so the tangential angle is

$\varphi = (n+1)\theta \pm \pi/2$.

(The sign here is positive if r and cos nθ have the same sign and negative otherwise.)

The unit tangent vector,

$\left(\frac{dr}{ds},\ r\frac{d\theta}{ds}\right)$,

has length one, so comparing the magnitude of the vectors on each side of the above equation gives

$\frac{ds}{d\theta} = r \cos^{-1} n\theta = a \cos^{-1+\tfrac{1}{n}} n\theta$.

In particular, the length of a single loop when $n>0$ is:

$a\int_{-\tfrac{\pi}{2n}}^{\tfrac{\pi}{2n}} \cos^{-1+\tfrac{1}{n}} n\theta\ d\theta$

The curvature is given by

$\frac{d\varphi}{ds} = (n+1)\frac{d\theta}{ds} = \frac{n+1}{a} \cos^{1-\tfrac{1}{n}} n\theta$.

Properties

The inverse of a sinusoidal spiral with respect to a circle with center at the origin is another sinusoidal spiral whose value of n is the negative of the original curve's value of n. For example, the inverse of the lemniscate of Bernoulli is a hyperbola.

The isoptic, pedal and negative pedal of a sinusoidal spiral are different sinusoidal spirals.

One path of a particle moving according to a central force proportional to a power of r is a sinusoidal spiral.

When n is an integer, and n points are arranged regularly on a circle of radius a, then the set of points so that the geometric mean of the distances from the point to the n points is a sinusoidal spiral. In this case the sinusoidal spiral is a polynomial lemniscate

References

• Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Spiral" p. 213–214
• "Sinusoidal spiral" at www.2dcurves.com
• "Sinusoidal Spirals" at The MacTutor History of Mathematics
• "Spirale Sinusoïdale" at Encyclopédie des Formes Mathématiques Remarquables
• Weisstein, Eric W., "Sinusoidal Spiral", MathWorld.